Conditional problem for objective probability.

*(English)*Zbl 1274.60013Summary: The marginal problem consists in finding a joint distribution whose marginals are equal to the given less-dimensional distributions. Let’s generalize the problem so that there are given not only less-dimensional distributions but also conditional probabilities.

It is necessary to distinguish between objective (Kolmogorov) probability and subjective (de Finetti) approach. In the latter, the coherence problem incorporates both probabilities and conditional probabilities in a unified framework. Different algorithms available for its solution are described, e.g., in [A. Gilio and S. Ingrassia, “Geometrical aspects in checking coherence of probability assessments”, in: Proceedings of the 6th international conference on information processing and management of uncertainty in knowledge-based systems (IPMU’96), Granada, Spain. 55–59 (1996); G. Coletti and R. Scozzafava, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4, No. 2, 103–127 (1996; Zbl 1232.03010)]. In the context of the former approach, it is shown that it is possible to split the task into solving the marginal problem independently and to subsequent solving the pure conditional problem as certain type of optimization.

First, an algorithm (conditional problem) that generates a distribution whose conditional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heuristical. Due to the computational complexity, it is efficient for small size problems, e.g., five dichotomical variables. Secondly, a method is mentioned how to unite marginal and conditional problem to a more general consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for a limited number of variables and conditionals. The described approach makes it possible to integrate additional knowledge, contained, e.g., in an empirical distribution, in the solution of the consistency problem.

It is necessary to distinguish between objective (Kolmogorov) probability and subjective (de Finetti) approach. In the latter, the coherence problem incorporates both probabilities and conditional probabilities in a unified framework. Different algorithms available for its solution are described, e.g., in [A. Gilio and S. Ingrassia, “Geometrical aspects in checking coherence of probability assessments”, in: Proceedings of the 6th international conference on information processing and management of uncertainty in knowledge-based systems (IPMU’96), Granada, Spain. 55–59 (1996); G. Coletti and R. Scozzafava, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4, No. 2, 103–127 (1996; Zbl 1232.03010)]. In the context of the former approach, it is shown that it is possible to split the task into solving the marginal problem independently and to subsequent solving the pure conditional problem as certain type of optimization.

First, an algorithm (conditional problem) that generates a distribution whose conditional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heuristical. Due to the computational complexity, it is efficient for small size problems, e.g., five dichotomical variables. Secondly, a method is mentioned how to unite marginal and conditional problem to a more general consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for a limited number of variables and conditionals. The described approach makes it possible to integrate additional knowledge, contained, e.g., in an empirical distribution, in the solution of the consistency problem.

##### MSC:

60A99 | Foundations of probability theory |

65C50 | Other computational problems in probability (MSC2010) |

##### Citations:

Zbl 1232.03010
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##### References:

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